On the Enumerative Geometry of Pascal's Hexagram

2023/03/18 by Jaydeep Chipalkatti, Chipalkatti, Jaydeep
Computer Science · Mathematics · #14N05 #14N10 #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2303.10319

openalex publication_date 2023/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given six points A,B,C,D,E,F on a nonsingular conic in the complex projective plane, Pascal's theorem says that the three intersection points AE ∩ BF, BD ∩ CE, AD ∩ CF are collinear. The line containing them is called a pascal, and we get altogether 60 such lines by permuting the points. In this paper, we consider the enumerative problem of finding the number of sextuples (A, B, …, F) which correspond to three pre-specified pascals. We use computational techniques in commutative algebra to solve this problem in all cases. The results are tabulated using the so-called 'dual' notation for pascals, which is based upon the outer automorphism of S₆.

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